August 22, 2014 | 6

Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter

I’ll be teaching a math history class for the first time this semester. I’m excited to be teaching it, but I’ve noticed that preparing for this class has been very different from preparing for other classes I’ve taught, which have all been math content courses.

I know how to teach a math content course. I don’t mean that I don’t have a lot to learn about teaching, but that I basically know what I want students to be able to do at the end of a math class, and that knowledge guides my teaching from day one. But when I first started preparing for my math history class, I wasn’t really sure how to start because I didn’t know where I wanted my students to finish.

At the end of a math content course, there are certain skills my students need to have. In calculus, they need to know what a derivative is, understand the chain rule, find the integral of a function. They’ll probably need to be able to use some of these skills in their next math classes.

Of course, it’s not just about content. I want my students to learn what qualifies as a convincing argument, persevere at problems, and communicate clearly about their thinking. Whether they continue taking math classes or not, those skills will help them in the future.

But it wasn’t quite as obvious to me what my students should get out of a math history class. It’s definitely not a list of when and where different mathematicians lived, or what theorems they proved. It’s not the life stories of famous mathematicians. Of course, I want it to help them develop their critical thinking and research skills, but that is too vague to be useful. Pretty much every college class should develop critical thinking skills.

A few facts about my class helped me think about what my goals should be. At my school, math history is a class in the math department, so it is probably more focused on mathematics than the same class would be if it were in the history department. It has a prerequisite of at least calculus I, and most of the students who enroll are majoring in math, engineering, or computer science. The course fulfills an upper-level writing and communication credit for the university, which means I have to make sure it satisfies certain requirements about how much writing students do.

I decided to let mathematics topics guide the course. Instead of presenting an overview of all of mathematics happening at one time in one place, my class will be somewhat modular. In each section, we’ll start early in history and follow the subject’s development over time. Mathematically, I’ll be focusing on number systems and number theory, the development of non-Euclidean geometry, and calculus. But I think more than in content courses, the exact topics I’m choosing to focus on are secondary. My students could get what I want them to get out of the class with a lot of different choices of topics.

If my students only get one thing out of this class, I want them to see that mathematics did not spring from the head of mathematicians fully-formed in little definition-theorem-proof packages. It was (and is) invented and discovered in bits and pieces by many people through creative processes, and people are still creatively inventing and discovering math.

I will be focusing on original sources as much as possible. I want my students to get practice reading unfamiliar mathematics and putting themselves in the shoes of the people who first wrote about it. By the end of the class, I my students should have experience reading original sources carefully and finding trusted secondary sources to illuminate the original sources if necessary. They should be able to do computations using some historical techniques and understand how those techniques relate to the modern ones they’re more familiar with.

As far as the writing component of the course is concerned, my main goal is for students to write clearly, correctly, and creatively to the audience that they intend to read a paper they write or listen to a presentation they deliver.

If you’ve taught a math history class, I’d be happy to hear what your goals for the course were and what you think about mine.

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Perhaps this is already built into your course and I hardly need mention it, but what I find most interesting in math history are the disputes, sometimes heated, over ideas between well-established proponents… over calculus, limits, infinitesimals, negative and complex numbers, infinity, Platonism, set theory, etc. (even more modern-day disputes over what math is and how to teach it). The incremental, steady evolution of math that lay people perceive as its history is rather boring (I think), but the disagreements are fascinating and revealing.

Link to thisI haven’t taught history of math, but I did take it, a rather mundane version that mostly recounted the highlights of the lives of a handful of well-known figures. From this experience, I have two suggestions: 1) Spend at least a bit of time locating the events and people you study in time and place. This challenged my (20-year-old) naive notion that most modern thinking happened in the 20th century! I found it quite illuminating to consider the contexts in which great math was invented. 2) Emphasize that math is not something you do, it’s a way of looking at the world. Point out that the people you study stepped outside the usual perspective to see events in (usually) radically different ways. Have fun with this class!

Link to thisI teach a history of math class as an elective at an advanced high school for math and science students. Many of them take calc 1 and 2 as sophomores and multivariable as juniors, so that leaves senior year free for a lot of electives.

I decided to focus on the 1830-1930 period as being the one that was most interesting and maybe most important. We read “Mathematics: the Loss of Certainty” by Morris Kline as a sort of textbook, going into more depth on certain topics as we go. I also spend a lot of time on the development of non-Euclidean geometry and on the controversies in dealing with infinity in analysis. But I use the second half of Kline’s book to focus a lot on the development of set theory, uncountability, formal axiomitizations, the paradoxes of set theory, Peano arithmetic, and the crises in the foundations of mathematics. We finish the course with a proof of Godel’s incompleteness theorem.

It works pretty well and I’m about to start teaching the class for the 4th time. You can find me on twitter @wrose31 if you want to talk more about what I expect them to know by the end of the course and how I test it.

Good luck!

Link to thisI find that the most compelling aspects of the course I teach (at least for me, and I hope for the students) are foundational questions. The students seem to enjoy swimming in philosophically muddy waters.

For example, this is really the only course in our curriculum (this is at Westminster College in SLC, UT) where we get to talk a lot about axiomatic systems. This raises all sorts of interesting philosophics questions about Platonism, formalism, and other ontological ideas. It’s a recurring theme throughout the course. It comes up right at the beginning: the early history of math in ancient civilizations raises questions about differences between “pure” math and “applied” math. It comes up, naturally, in Greek geometry. It comes up when talking about infinitessimals in calculus. (This is where we talk about Wigner’s “Unreasonable Effectiveness of Mathematics”.) And it comes up at the end in non-Euclidean geometry and logic (Godel, etc.), especially in the context of using models to check the relative consistency of axiomatic systems.

I also try to be very intentional about the role of women, minorities, and non-Western cultures (the presence or unfortunate lack thereof) throughout the whole course.

Link to thisThanks, everyone, for your helpful comments and suggestions. I know I can’t do everything this semester, but you’ve given me a lot to think about. I hadn’t really thought about including other historical context in the class. That sounds like a great idea.

Link to thisOne of my favorite stories is why an hour has 60 minutes, because it happens to be a crash course on the history of science & math over the entirety of human civilization. I discovered it while reading “An Introduction to the History of Algebra” by Jacques Sesiano (which I highly recommend).

If you’re interested, I wrote the 60-minute story as an article for LiveScience. I probably can’t link it here, but if you google “Keeping Time: Why 60 Minutes?” you should find it pretty quick.

Love the blog. Looking forward to what you have to say about math history.

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